Spatiality of Derivations on the Algebra of τ-Compact Operators

This paper is devoted to derivations on the algebra S (0)(M, tau) of all tau-compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace tau. The main result asserts that every t (tau) -continuous derivation is spatial and implemented by a tau-measurable operator affiliated with M, where t (tau) denotes the measure topology on S (0)(M, tau). We also show the automatic t (tau) -continuity of all derivations on S (0)(M, tau) for properly infinite von Neumann algebras M. Thus in the properly infinite case the condition of t (tau) -continuity of the derivation is redundant for its spatiality.